# on an interval

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##### 1: 1.4 Calculus of One Variable

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►If $f({x}_{1})\le f({x}_{2})$ for every pair ${x}_{1}$, ${x}_{2}$ in an interval
$I$ such that $$, then $f(x)$ is

*nondecreasing*on $I$. … ►If also $f(x)$ is continuous on the right at $x=a$, and continuous on the left at $x=b$, then $f(x)$ is*continuous on the interval*$[a,b]$, and we write $f(x)\in C[a,b]$. … ►If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is*piecewise*(or*sectionally*) continuous on $I$. … ►If ${f}^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in {C}^{n}(I)$. …##### 2: 26.2 Basic Definitions

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►If, for example, a permutation of the integers 1 through 6 is denoted by $256413$, then the cycles are $(1,2,5)$, $(3,6)$, and $(4)$.
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►Unless otherwise specified, it consists of horizontal segments corresponding to the vector $(1,0)$ and vertical segments corresponding to the vector $(0,1)$.
For an example see Figure 26.9.2.
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►A

*partition of a nonnegative integer*$n$ is an unordered collection of positive integers whose sum is $n$. As an example, $\{1,1,1,2,4,4\}$ is a partition of 13. …##### 3: 28.17 Stability as $x\to \pm \mathrm{\infty}$

##### 4: 18.36 Miscellaneous Polynomials

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►These are OP’s on the interval
$(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials.
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##### 5: Bibliography K

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Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type.
ACM Trans. Math. Software 22 (4), pp. 385–392.
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##### 6: 18.2 General Orthogonal Polynomials

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►It is assumed throughout this chapter that for each polynomial ${p}_{n}(x)$ that is orthogonal on an open interval
$(a,b)$ the variable $x$ is confined to the closure of $(a,b)$

*unless indicated otherwise.*(However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … ►More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by a positive measure $d\alpha (x)$, where $\alpha (x)$ is a bounded nondecreasing function on the closure of $(a,b)$ with an infinite number of points of increase, and such that $$ for all $n$. … ►All $n$ zeros of an OP ${p}_{n}(x)$ are simple, and they are located in the interval of orthogonality $(a,b)$. …##### 7: 10.23 Sums

##### 8: 1.6 Vectors and Vector-Valued Functions

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$\mathbf{c}(t)=(x(t),y(t),z(t))$, with $t$ ranging over an interval and $x(t),y(t),z(t)$ differentiable, defines a

*path*. … ►with $(u,v)\in D$, an open set in the plane. …##### 9: 2.3 Integrals of a Real Variable

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$k$ ($\le \mathrm{\infty}$) and $\lambda $ are positive constants, $\alpha $ is a variable parameter in an interval
${\alpha}_{1}\le \alpha \le {\alpha}_{2}$ with ${\alpha}_{1}\le 0$ and $$, and $x$ is a large positive parameter.
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